A Better Butterfly Theorem

The following generalization of the Butterfly Problem has been pointed out to me by Qiu Fawen, a Chinese teacher who discovered the result with his students in 1997. The Chinese version was at some time available at qiusir.com. I had to do some guess work to figure out what it was about.

According to Leon Bankoff (Mathematics Magazine, Volume 60, No. 4, October 1987, pp. 195-210) the appellation The Butterfly made its first appearance in the Solutions section of the American Mathematical Monthly in the February 1944 issue. Since the diagram of the theorem (with two wings on each side) established by Qiu Fawen and his students gives a more realistic depiction of a butterfly, I suggest to call the statement A Better Butterfly Theorem which might be interpreted as both A "Better Butterfly" Theorem and A Better "Butterfly Theorem", the former being my preference.

A Better Butterfly Theorem

Let there be two concentric circles with the common center O. A line crosses the two circles at points P, Q and P', Q', M being the common midpoint of PQ and P'Q'. Through M, draw two lines AA'B'B and CC'D'D and connect AD', A'D, BC', and B'C. (This is the Butterfly.) Let X, Y, Z, W be the points of intersection of PP'Q'Q with AD', B'C, A'D, and BC', respectively. Then

(1) 1/MX + 1/MZ = 1/MY + 1/MW.

(Since X coincides with Z and Y with W when the two circles coalesce into one, The Butterfly Theorem is an immediate consequence of A Better Butterfly Theorem.)

The proof depends on the following

Lemma

Let in ΔRST, RU be a cevian through vertex R. Introduce angles a = ∠SRU and b = ∠URT. Then

(2) sin(a + b)/RU = sin(a)/RT + sin(b)/RS.

The proof follows from the fact that Area( ΔRST) = Area( ΔRSU) + Area( ΔRUT) by a consistent application of the sine formula for the area of a triangle. (For example, Area( ΔRST) = RS·RT·sin(a + b)/2.)

Proof of the theorem

We apply Lemma to triangles AMD', A'MD, B'MC, and BMC':

(3) sin(a + b)/MX = sin(a)/MD' + sin(b)/MA,
sin(a + b)/MZ = sin(a)/MD + sin(b)/MA',
sin(a + b)/MY = sin(a)/MC + sin(b)/MB',
sin(a + b)/MW = sin(a)/MC' + sin(b)/MB.

In view of (3), (1) will follow from

(4) sin(a)/MD' + sin(b)/MA + sin(a)/MD + sin(b)/MA' =
      sin(a)/MC + sin(b)/MB' + sin(a)/MC' + sin(b)/MB,

or, which is the same, from

(5) sin(b)(1/MA - 1/MB) + sin(b)(1/MA' - 1/MB') =
      sin(a)(1/MC - 1/MD) + sin(a)(1/MC' - 1/MD').

Now, drop perpendiculars OM1 and OM2 from O onto AA'B'B and CC'D'D. M1 is the midpoint of both AB and A'B', whereas M2 is the midpoint of CD and C'D'. Obviously,

(6) MB - MA = MB' - MA' = 2·OM1 = 2·OM·sin(a) and
MD - MC = MD' - MC' = 2·OM2 = 2·OM·sin(b).

With (6) in mind, (5) is equivalent to

(7) 1/MA·MB + 1/MA'·MB' = 1/MC·MD + 1/MC'·MD'.

However, as is well known, MA·MB = MC·MD and MA'·MB' = MC'·MD'. Therefore, (7) is true, as is (5), which in turn, implies (1).

Butterfly Theorem and Variants

  1. Butterfly theorem
  2. 2N-Wing Butterfly Theorem
  3. Better Butterfly Theorem
  4. Butterflies in Ellipse
  5. Butterflies in Hyperbola
  6. Butterflies in Quadrilaterals and Elsewhere
  7. Pinning Butterfly on Radical Axes
  8. Shearing Butterflies in Quadrilaterals
  9. The Plain Butterfly Theorem
  10. Two Butterflies Theorem
  11. Two Butterflies Theorem II
  12. Two Butterflies Theorem III
  13. Algebraic proof of the theorem of butterflies in quadrilaterals
  14. William Wallace's Proof of the Butterfly Theorem
  15. Butterfly theorem, a Projective Proof
  16. Areal Butterflies
  17. Butterflies in Similar Co-axial Conics
  18. Butterfly Trigonometry
  19. Butterfly in Kite
  20. Butterfly with Menelaus
  21. William Wallace's 1803 Statement of the Butterfly Theorem
  22. Butterfly in Inscriptible Quadrilateral
  23. Camouflaged Butterfly
  24. General Butterfly in Pictures
  25. Butterfly via Ceva
  26. Butterfly via the Scale Factor of the Wings
  27. Butterfly by Midline
  28. Stathis Koutras' Butterfly
  29. The Lepidoptera of the Circles
  30. The Lepidoptera of the Quadrilateral
  31. The Lepidoptera of the Quadrilateral II
  32. The Lepidoptera of the Triangle
  33. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  34. Two Butterfly Theorems by Sidney Kung

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